Taylor expansion/R/One variable/Introduction/Section
So far, we have only considered power series of the form . Now we allow that the variable may be replaced by a "shifted variable“ , in order to study the local behavior in the expansion point . Convergence means, in this case, that some exists, such that for
![](http://upload.wikimedia.org/wikipedia/commons/thumb/5/53/Taylor_Brook_Goupy_NPG.jpg/250px-Taylor_Brook_Goupy_NPG.jpg)
the series converges. In this situation, the function, presented by the power series, is again differentiable, and its derivative is given as in fact. For a convergent power series
the polynomials yield polynomial approximations for the function in the point . Moreover, the function is arbitrarily often differentiable in , and the higher derivatives in the point can be read of from the power series directly, namely
We consider now the question whether we can find, starting with a differentiable function of sufficiently high order, approximating polynomials (or a power series). This is the content of the Taylor expansion.
Let denote an interval,
an -times differentiable function, and . Then
So
is the constant approximation,
is the linear approximation,
is the quadratic approximation,
is the approximation of degree , etc. The Taylor polynomial of degree is the (uniquely determined) polynomial of degree with the property that its derivatives and the derivatives of at coincide up to order .
Let denote a real interval,
an -times differentiable function, and an inner point of the interval. Then for every point , there exists some such that
and .
Proof
![](http://upload.wikimedia.org/wikipedia/commons/thumb/2/21/Sintay.svg/250px-Sintay.svg.png)
Suppose that is a bounded closed interval,
is an -times continuously differentiable function, an inner point, and . Then, between and the -th Taylor polynomial, we have the estimate